Kelvin–Stokes theorem

The Kelvin–Stokes theorem, named after Lord Kelvin and George Stokes, also known as the Stokes' theorem, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on R 3 {displaystyle mathbb {R} ^{3}} . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface.Definition 2-1 (Irrotational field). A smooth vector field, F on an open U ⊆ R3 is called a irrotational if ∇ × F = 0.Let U ⊆ R3 be an open subset with a Lamellar vector field F, and piecewise smooth loops c0, c1: → U. If there is a function H: × → U such that ∫ c 0 F d c 0 = ∫ c 1 F d c 1 {displaystyle int _{c_{0}}mathbf {F} ,dc_{0}=int _{c_{1}}mathbf {F} ,dc_{1}} Lemma 2-2. Let U ⊆ R3 be an open subset, with a Lamellar vector field F and a piecewise smooth loop c0: → U. Fix a point p ∈ U, if there is a homotopy (tube-like-homotopy) H: × → U such that ∫ c 0 F d c 0 = 0 {displaystyle int _{c_{0}}mathbf {F} ,dc_{0}=0} Definition 2-2 (Simply Connected Space). Let M ⊆ Rn be non-empty, connected and path-connected. M is called simply connected if and only if for any continuous loop, c: → M there exists H: × → M such thatTheorem 2-2. Let U ⊆ R3 be a simply connected and open with an irrotational vector field F. For all piecewise smooth loops, c: → U we have: ∫ c 0 F d c 0 = 0 {displaystyle int _{c_{0}}mathbf {F} ,dc_{0}=0} Definition 3-1 (Singular 2-cube) Set D = × ⊆ R2 and let U be a non-empty open subset of R3. The image of D under a piecewise smooth map ψ: D → U is called a singular 2-cube. Moreover, we define the notarization map of DLemma 3-1 (Notarization map of singular two cube). Let D be a singular 2-cube with map ψ and U ⊆ R3 open and non-empty. Suppose the image of I × I under a piecewise smooth map φ ∘ θ D , S ~ := φ ∘ θ D ( I × I ) {displaystyle varphi circ heta _{D},{widetilde {S}}:=varphi circ heta _{D}(I imes I)} be a singular 2-cube. If F is a smooth vector field on U we have:Definition 3-2 (Cube subdivisionable sphere). (see Iwahori p. 399) A non-empty subset S ⊆ R3 is said to be a 'Cube subdivisionable sphere' when there are at least one Indexed family of singular 2-cubeDefinitions 3-3 (Boundary of a Cube Subdivision Sphere). (see Iwahori p. 399) Let S ⊆ R3 be a cube subdivisionable sphere with cube subdivision:Lemma 3-4. (see Iwahori p. 399) Let S ⊆ R3 be a cube subdivisionable sphere with cube subdivisions:Definitions 3-5 (Boundary of Surface). (see Iwahori p399) Let S ⊆ R3 be a cube subdivisionable sphere and, { ( I 2 , φ λ , S λ ) } λ ∈ Λ {displaystyle {(I^{2},varphi _{lambda },S_{lambda })}_{lambda in Lambda }} , thenDefinition (Homotopy and Homotope). Suppose Z and W are topological spaces, with continuous maps f0, f1: Z → W.Definition (tube-like homotopy and tube-homotope). Suppose c0, c1 satisfy the following:Definition (Joint of paths). Let M be a topological space and α: → M, β: → M, be two paths on M. If α and β satisfy α(b1) = β(a2) then we can join them at this common point to produce new curve α ⊕ β : → M defined by:Definition (Backward of curve). Let M be a topological space and α : → M,be path on M. We can define backward thereof, ⊖ {displaystyle ominus } α : → M by:And, given two curves on M, α: → M, β: → M, which satisfy α(b1 = β(b2) (that means α(b1) = ⊖ {displaystyle ominus } β(a2), we can define α ⊖ β {displaystyle alpha ominus eta } as following manner. The Kelvin–Stokes theorem, named after Lord Kelvin and George Stokes, also known as the Stokes' theorem, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on R 3 {displaystyle mathbb {R} ^{3}} . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. If a vector field A = ( P ( x , y , z ) , Q ( x , y , z ) , R ( x , y , z ) ) {displaystyle mathbf {A} =(P(x,y,z),Q(x,y,z),R(x,y,z))} is defined in a region with smooth surface Σ {displaystyle Sigma } and has first order continuous partial derivatives then: where ∂ Σ {displaystyle partial Sigma } is boundary of region with smooth surface Σ {displaystyle Sigma } . The Kelvin–Stokes theorem is a special case of the “generalized Stokes' theorem.” In particular, a vector field on R 3 {displaystyle mathbb {R} ^{3}} can be considered as a 1-form in which case curl is the exterior derivative. Let γ: → R2 be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that γ divides R2 into two components, a compact one and another that is non-compact. Let D denote the compact part that is bounded by γ and suppose ψ: D → R3 is smooth, with S := ψ(D). If Γ is the space curve defined by Γ(t) = ψ(γ(t)) and F is a smooth vector field on R3, then The proof of the theorem consists of 4 steps. We assume Green's theorem, so what is of concern is how to boil down the three-dimensional complicated problem (Kelvin–Stokes theorem) to a two-dimensional rudimentary problem (Green's theorem). When proving this theorem, mathematicians normally use the differential form. The 'pull-back of a differential form' is a very powerful tool for this situation, but learning differential forms requires substantial background knowledge. So, the proof below does not require knowledge of differential forms, and may be helpful for understanding the notion of differential forms.